What is the best way...  

And I realize
As a noob, I probably should simply read through the threads and see if my question was answered elsewhere. I have been on other types of forums for years and know how annoying new collectors can be when they ask obvious questions. ;-/
Sorry.

Joshua,
underneath the armrests you will be able to unscrew them and take them off. You will then expose the metal armrest plates which again you can simply unscrew.
Once you have done this you will be able to physically inspect the shockmounts.
You will be able to check the condition of the rubber mounts and see as best you can any weakness to the adhesive holding them on.

Thanks 😉
Appreciate the feedback. Can the original mounts be reglued/reinforced if I find them to have issues? Meaning if the rubber itself is fine but the glue connection might be weakening?
I am only asking as personally I would like to keep the integrity of the original chair and the idea of a new part bothers me slightly.
At the same time, I would prefer to have it be safe, a failure and potential damage to the Rosewood shell certainly is the main concern.

I'd do a close visual inspection
as H.Moon suggests. Look for evidence of any cracking, hardening, or separation. If the arm pads have never been off, apply firm, steady pressure after removing the two screws. They will eventually release.
While the chair is assembled, feel for excessive movement while someone is sitting in the chair by putting your fingertips in there, if they fit.
You might consider planning for replacement at some point soon, either by HM or Hume. I think you're right to be concerned, but, if all seems fine, I'd go ahead and use the chair at will. Just avoid flopping yourself into it, like my teenagers like to do with mine, and hide the chair when your overweight brother-in-law comes to visit.
I don't think competent shock mount replacement should devalue your 670 in any way. In fact, I'd almost expect to see it on a well cared for example of that vintage.

.
I got rid of an old Aalto 31 because I was too scared to sit in it, and I'm only 60 kilos or so - I reckon an Eames 670 will be much more confidence inspiring, despite not knowing how many corpulent Americans have been there 😉
On a similar note there is an Marcel Breuer Isokon chair on Ebay at mo - I saw it when it first came out at a market - absolutely beautiful, very delicate, very rare......and absolutely riddled, riddled, with woodworm. It must be 75% air inside the plywood judging by the amount of exit holes I saw. Nice to look at but I doubt it would take even a small childs' weight.

One other tip I'd suggest...
One other tip I'd suggest (which helped me locate an impending failure on my 670) is that once you've taken the armrests off, take off the back first, but leave the metal brackets on. You can use the brackets to apply force in various directions to the shock mount and, while doing this, see if the mounts appear to be separating from the shell at any spot. One on of mine, I tried this and found the mount had separated about 1/3 of the way from the shell!

Honestly, I would take a...
Honestly, I would take a vintage 670 with professionally replaced shock mounts over an equivalent vintage example with original shock mounts.

I agree
I agree, they all need to be replaced at some point and I would prefer that they were professionally repaired. My good friend at Herman Miller said that the shock mount repairs today are both mechanically attached with some small screws and glued. On another note, I also heard that a larger, oversized version of the 670 Lounge Chair is in the works at Herman Miller

I think...
Maybe that is what concerns me. It is just my personal neuroticness ( is that even a word) but the idea of any screws or non original parts in the chair sort of concern me.
It is not that I am going to sell it, but maybe I am a naive purist noob....I just would prefer to not drill anything into the shell period.
Thanks though for the varied feedback, it is very helpful.

Well
It seems the mounts will have to be replaced. I had the chair checked and the mounts were re glued at some point in their life and , it is obvious not by someone professional. I plan to send them to herman Miller and get them done. At least this will allow me to sit in it once in a while without cringing. I am sad I could not keep the original mounts, but I suppose most of them fail and better to replace it properly now than damage the shell later and kick myself.
I just have to wonder outside of the hardcore collectors, how much the new Herman Miller mounts affect the value of the chair. Maybe I am over thinking it.
Charles would probably say "replace the damn mounts" already. 😉

Putting brand new professionally fitted shock mounts on your vintage chair will not effect its value, in many cases people will find it reassuring.
Think of it as putting a new set of tyres on your vintage car. Sit back and enjoy your chair Joshua.

or like replacing a timing belt in your car...
If that fails,boy what an expensive mess